Properties

Label 2.61.az_kl
Base Field $\F_{61}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{61}$
Dimension:  $2$
L-polynomial:  $1 - 25 x + 271 x^{2} - 1525 x^{3} + 3721 x^{4}$
Frobenius angles:  $\pm0.0746782053317$, $\pm0.283932848163$
Angle rank:  $2$ (numerical)
Number field:  4.0.1639109.1
Galois group:  $D_{4}$
Jacobians:  5

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 5 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2443 13541549 51548582575 191740222452149 713336156254883728 2654331215399083968125 9876822194764167581661583 36751691963860975798195584869 136753054580313921280528613143675 508858111517766385885362591180873984

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 37 3639 227107 13848219 844588302 51520029663 3142739546347 191707303124019 11694146241606517 713342914323716854

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
The endomorphism algebra of this simple isogeny class is 4.0.1639109.1.
All geometric endomorphisms are defined over $\F_{61}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.61.z_kl$2$(not in LMFDB)